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{\bf E. S. Egge, J. Haglund, K. Killpatrick and D. Kremer}
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{\bf A Schr\"oder Generalization of Haglund's Statistic on Catalan Paths}
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Garsia and Haiman ({\it J. Algebraic. Combin.} $\bf5$ $(1996)$,
$191-244$) conjectured that a certain sum $C_n(q,t)$ of rational
functions in $q,t$ reduces to a polynomial in $q,t$ with nonnegative
integral coefficients. Haglund later discovered ({\it Adv. Math.}, in
press), and with Garsia proved ({\it Proc. Nat. Acad. Sci.} ${\bf98}$ $(2001)$, $4313-4316$) the refined conjecture $C_n(q,t) = \sum
q^{{\rm area}}t^{{\rm bounce}}$. Here the sum is over all Catalan
lattice paths and ${\rm area}$ and ${\rm bounce}$ have simple
descriptions in terms of the path. In this article we give an
extension of $({\rm area},{\rm bounce})$ to Schr\"oder lattice
paths, and introduce polynomials defined by summing
$q^{{\rm area}}t^{{\rm bounce}}$ over certain sets of Schr\"oder
paths. We derive recurrences and special values for these
polynomials, and conjecture they are symmetric in $q,t$. We also
describe a much stronger conjecture involving rational functions in
$q,t$ and the $\nabla$ operator from the theory of Macdonald symmetric
functions.
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